If Mao and Attlee could not get rid of it, don't expect much progress in the foreseeable future. I'm talking class. The ability of China's Communist party to reinvent itself as a version of the Confucian elite it purported to eliminate - a Chinese Animal Farm - and the capacity of Britain's private-school system to carry on supplying most of Britain's elite are two sides of the same coin. Class, like death and taxes, is always with us.

The resourcefulness of class in its resistance to social levelling, whether its roots are parliamentary or revolutionary, is extraordinary. A classless society has proved a chimera. Set our sights differently, though, and the achievement of a pluralist and fair society could and should be possible.

Yet even that is elusive. Last week saw the Sutton Trust once again capture the headlines with an intriguing study on who dominates the media. The grip of the ex-independent-school boy and girl had got tighter, it reported, since a similar study in 1986. They now occupy more than half the senior editorial and opinion-forming roles. It's a similar pattern in business, the judiciary and the financial system.

And it matters. One example sticks in my gullet - the space our media give to criticising universities which attempt a more equitable representation of children from state schools while showing no interest in the collapse of Britain's apprenticeship system. Maybe this is because at their editorial conferences no one is trying to get a child an apprenticeship. But they are constantly alert to any hint that top universities might be indulging in 'social engineering' to the disadvantage of their children.

I never expect to see Oxford or Cambridge reduce admissions from independent schools to fairer levels. That is not their social function. Their role is analogous to China's Communist party schools, France's Ecole Nationale d'Administration or the US's Ivy League. They are gatekeepers to the elite, recruiting students whose background and bearing suggest they will make good elite members. They come disproportionately from independent schools, just as in China they come disproportionately from the families of senior officials.

Naturally, all have exceptionally good formal exam results, so that suggestions of social elitism can be firmly rebutted at Oxbridge's high tables or at China's equivalent. The evidence, however, speaks otherwise.

This is the ever-present reality. What is intriguing is the recognition that it is getting worse and that, in Britain, the drivers of social mobility are weakening. Research by Jo Brandon, Paul Gregg and Steve Machin for the London School of Economics shows that while the proportion of children from the bottom 20 per cent gaining degrees has only increased fractionally since the 1970s, at the same time, the proportion from the top 20 per cent achieving degrees has more than doubled. The expansion of higher education has massively benefited the children of better-off families.

The British middle class is operating a national, de facto closed shop, very much like the old print unions or dock workers. By ensuring that it is largely their children who win degrees, they keep hold of the entry tickets to high-status jobs. And independent schools are a crucial part of the system.

But, as the Sutton Trust report accepts, school and university are necessary preconditions for access to high-status jobs, but alone they are not sufficient. To succeed in the media, you have to have the means to live in expensive London while receiving dirt-poor initial wages; you need connections to get a first job in an industry where entry is notoriously unmeritocratic; and you need the inner self-confidence and external capacity to present yourself well. The privately educated score better on all counts against their state- school rivals.

Why are these effects greater now than 20 years ago? One factor is that the huge expansion of universities has placed a higher premium on non-formal means of selecting candidates; the subtle accoutrements of class matter more. Another factor is that London has become more expensive and the growth in starter salaries has not kept pace. Having parents who can support you early in your career is more crucial. London house prices prop up the middle class's closed shop as effectively as independent schools.

But perhaps the subtlest and most important reason for decreased mobility is that inequality has become culturally acceptable. Twenty years ago, a large part of the liberal middle class regarded it as a moral imperative to send their children to state schools; that attitude has nearly evaporated. The culture that patronises chavs and Wayne Rooney's girlfriend also indulges independent schools.

This world view has emerged not only with the celebration, almost fetishisation, of choice and individualism, but also from the collapse of vigorous working-class institutions that might have kept the wider culture more honest. Inevitably, we have to rely on an old Etonian Tory baronet - Ferdinand Mount (related to David Cameron's mother) - to get an insight into the dynamics of class seen from the bottom not the top.

In Mind the Gap, a book that should have received much more attention when it was published a couple of years ago, he celebrates the 19th-century development of great working-class institutions from trade unions to mutual building societies and working men's clubs, deploring their disastrous disintegration over the past 50 years. Working-class culture and leadership have been eviscerated.

Mount's view is that Britain cannot build a more vigorously pluralist and fairer society without stronger working-class institutions. I agree. If we cannot achieve a classless society, a pluralist one may be within our grasp. But it will need modern trade unions and new institutions of working-class self-help.

A pluralist society would not mean abolishing independent schools, but they cannot continue to dominate entire professions. Britain should be more than a closed shop for its middle class. On current trends, that is what it will become within a further 20 years.